The universe expands with a Hubble velocity [1]}, and the expansion of the universe affects the
dynamics of objects with gravitational forces.
Simulations of galaxies with dark matter in an expanding universe have been performed for many decades. In
[2]}, [3]} the authors solved the dynamics of baryonic objects
by the Particle-Particle/Particle-Mess (PPPM) method [4]},
where each mass unit is treated as moving in the collective field of all others, and the Poison equation
for the PPPM grid is solved numerically. The dynamics are obtained for Zeldovich's adiabatic expansion of the Einstein-de Sitter universe [5]}.
Later, simulations with large scaled computer packages with PPPM [6]}, [7]} are with many billions of mass units.
The evolution of galaxies has also been obtained from hydrodynamical large scaled cosmological simulations [8]}, [9]}, [10]},
and with
co-evolving dark matter gas and stellar objects [11]}, [12]}. Cosmological simulations of galaxy formation is reviewed in [13]}.
